Could someone please check if the procedure for part a) is correct and I need a bit of help in regards to part b). I don't have the answer to refer to. Thank you for your time in advance!
An airport reports that on a stormy day, with wind speed between 60 and 70 kilometers per hour, 20% of the domestic flichts are cancelled, 30% are delayed and 50% are on time. Tomorrow 12 domestic flights are on schedule and the weather is expected to be stormy with average wind speed of 65 kilometer per hour.
a) Compute the probability that exactly 3 flights are cancelled.
So here we have Binomial Distribution with $n=12$ and $p=0.2$ Since we have only 2 possible outcomes as far as we are concerned for this case, either a flight is cancelled or a flight is not cancelled.
The formula for Binomial Distribution is as follows: $P(k)=\binom{n}{k}p^k(1-p)^{n-k}$ Hence we just plug in the values from above:
$P(3)=\binom{12}{3}0.2^3(1-0.2)^{12-3}=0.23622$
b) Determine the probability that less than 9 but at least 2 flights are punctual.
So for this probability space (again Binomial distribution), we have $n=12$ $p=0.5$. However, I am confused in regards to what to do here. Do I have to compute every single probability and then combine them? Aka P(2),P(3),P(4),P(5),P(6),(7),P(8)? That seems to be rather long and as if I am missing something here so any help would be much much appreciated! Thank you!
In part b) you could also calculate probability of the opposite event: probability that $9$ or more or less than $2$ flights are on time.
$$P_{complement} = P(0) + P(1) + P(9) + P(10) + P(11) + P(12)$$
Here you have to sum only $6$ probabilities and the probability of original event from question is:
$$P = 1 - P_{complement} = 1 - P(0) - P(1) - P(9) - P(10) - P(11) - P(12)$$
If this isn't an exam question you can also write a simple computer program that computes desired probabilities.
P.S. When I was learning probability and statistics first time I found writing computer programs that model problems from textbooks very insightful.